mathematics Article Risk Analysis through the Half-Normal Distribution Maria-Teresa Bosch-Badia 1 , Joan Montllor-Serrats 2, * and Maria-Antonia Tarrazon-Rodon 2 1 Deparment of Busines, Campus de Montilivi, Universitat de Girona, 17071 Girona, Spain; MariaTeresa.Bosch@udg.edu 2 Department of Business, Universitat Autonoma de Barcelona, 08193 Cerdanyola del Valles, Spain; MariaAntonia.Tarrazon@uab.cat * Correspondence: Joan.Montllor@uab.cat; Tel.: +34-93-581-1269  Received: 2 November 2020; Accepted: 13 November 2020; Published: 21 November 2020  Abstract: We study the applicability of the half-normal distribution to the probability–severity risk analysis traditionally performed through risk matrices and continuous probability–consequence diagrams (CPCDs). To this end, we develop a model that adapts the financial risk measures Value-at-Risk (VaR) and Conditional Value at Risk (CVaR) to risky scenarios that face only negative impacts. This model leads to three risk indicators: The Hazards Index-at-Risk (HIaR), the Expected Hazards Damage (EHD), and the Conditional HIaR (CHIaR). HIaR measures the expected highest hazards impact under a certain probability, while EHD consists of the expected impact that stems from truncating the half-normal distribution at the HIaR point. CHIaR, in turn, measures the expected damage in the case it exceeds the HIaR. Therefore, the Truncated Risk Model that we develop generates a measure for hazards expectations (EHD) and another measure for hazards surprises (CHIaR). Our analysis includes deduction of the mathematical functions that relate HIaR, EHD, and CHIaR to one another as well as the expected loss estimated by risk matrices. By extending the model to the generalised half-normal distribution, we incorporate a shape parameter into the model that can be interpreted as a hazard aversion coefficient. Keywords: half-normal distribution; generalised half-normal distribution; risk matrices; continuous probability–consequence diagrams 1. Introduction The probability–severity risk analysis usually estimates the risk levels of different scenarios through the product between the probability and the size of hazards impacts in the framework of risk matrices or continuous probability–consequence diagrams (CPCDs). The current research aims to go a step further by building up a model that adapts the financial risk measures Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), the latter also denominated expected short-fall to the probability–severity analysis by taking the half-normal distribution as its central methodological pillar. We call it the “Truncated Risk Model” (TR model) because it admits the substitution of the half-normal distribution by any other truncated probability distribution that fits the scenarios under study. In concordance with the contents of risk matrices, the indicators we present focus on downside risk strictly. Thus, they only capture negative impacts, although, for convenience, we express them in positive units. The uniqueness of the signs makes a truncated probability distribution the appropriate tool for expressing the random nature. Indeed, truncated probability distributions are widely used in risk analysis. In the field of finance, among many other works, Aharony et al. [1] applied a truncated normal distribution to corporate bankruptcies. Truncations are also central in the Collins and Gbur [2] study on limited liability through the lenses of safety-first rules. Ergashev et al. [3] used them in a paper on operational risk modelling. Chen et al. [4] employed the half-normal distribution in an Mathematics 2020, 8, 2080; doi:10.3390/math8112080 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2080 2 of 27 estimation of the general Bayesian VaR. De Roon and Karehnke [5] built a transformation of the half-normal distribution, the denominated smooth half-normal, that they apply to VaR and CVaR, among other indicators. Beyond the financial world, truncated distributions are frequently used also in the analysis of natural risk phenomena. Jawitz [6] studied truncated distributions from the perspective of hydrologic problems. Li et al. [7] presented an application to flood frequencies. Lasar and Dolsek [8] analysed seismic risk through truncated distributions. Other applications of truncated distributions can also be found in the fields of engineering [9] and biology [10], including medicine. In this respect, we apply the half-normal distribution for its direct connection with the usual presentations of VaR and CVaR through the normal distribution. An extension to the generalised half-normal distribution is also part of the TR model, allowing to complement the scale parameter of the half-normal case with a shape parameter. The generalised half-normal distribution (GHN), proposed by Cooray and Ananda [11], incorporates a shape parameter to the original half-normal distribution, adding flexibility to its analytical capacity. Pescim et al. [12] related the generalised half-normal to beta distribution. The empirical applications of these two papers have focused on lifetime and health data. Risk matrices are widely used in risk analysis despite their limitations. As known, risk matrices identify risk levels by combining the likelihood of hazards with the severity of their impacts, usually through a product. However, the design of risk matrices requires careful attention. The main limitation of risk matrices is that they cannot eliminate residual ambiguity in the classification of risks completely. Risk analysis literature has discussed the applicability of risk matrices and the guidelines for their design. Hopkin [13], in his handbook on risk management, dealt with the implementation of risk matrices in this field. Cox [14] reviewed the inconsistencies that risk matrices may present and developed an axiomatic system for their rational construction. Among the inconsistencies, Cox ([14], p. 507) pointed out that risk matrices may transgress the principle of translation invariance for coherent risk measures [15]. This transgression is not due to the risk indicator but to the structure of risk matrices. Departing from Cox’s axioms, Li et al. [16] built a method (the Sequential Updating Approach) for minimising wrong risk pairs in risk matrices. Aven [17] depicted the limitations of risk matrices, pointing out that they are a tool for describing risk but cannot be taken as a risk analysis method ([17] p. 143). The same author [18] pledged to broaden the plain likelihood-severity content of risk matrices by adding the knowledge dimension of risk. Levine [19] explored the improvements that logarithmic scales can introduce into risk matrices. Baybutt [20] underlined the importance of calibrating risk matrices and proposed a set of rules to avoid pitfalls in this process. The TR model built in this paper overcomes the limitations of risk matrices, and, at the same time, incorporates risk indicators that can be interpreted in line with the widely accepted risk measures VaR and CVaR but without being constrained to financial analysis. The ambiguities in rankings can be overcome by substituting the rectangular structure of risk matrices in the delimitation of risk categories by continuous iso-risk lines, namely lines that denote the same risk level. This change of indicators leads to a switch from risk matrices to continuous risk combinations. Continuous combinations have been less studied in the risk literature. An appropriate denomination for them, according to Duijm ([21], p. 29), is “continuous probability–consequence diagrams” (CPCDs). The continuous probability–consequence diagrams (CPCD) constitute an alternative to risk matrices, switching from the discrete scales of risk matrices to continuous scales based on iso-risk lines. Ale et al. [22] reviewed the origins and evolution of probability–consequence diagrams as an instrument of risk analysis. Duijm [21], after clarifying that risk matrices are part of probability–consequence diagrams ([21], p. 22), as a discrete version of them, developed an in-depth analysis of the comparison between risk matrices and CPCDs, namely between the discrete and continuous versions of probability–severity risk analysis. This author pointed out that, in CPCDs, “it is possible to discriminate between hazards that, in a discrete matrix, would be assigned to the same cell” ([21], p. 29). Thus, the incorporation of continuity is central for improving the analysis. Ni et al. [23] developed continuous divisions of risk matrices comparing the effects of different arithmetic combinations between probability and severity. Laine et al. [24] compared risk matrices and Mathematics 2020, 8, 2080 3 of 27 CPCDs in the framework of risk management of pollution at sea. This paper uses the denomination “probability–consequence diagrams” for their continuous version, i.e., for CPCDs, presenting them as a generalisation of risk matrices for continuous scales of measurement ([24], p. 89). The dominant risk estimator in risk matrices is the product between the probability of the hazard and the size of its impact. This measure is extended to CPCDs. Estimating the risk level in this way reduces the estimation of each hazard to single values of its probability and impact size, instead of representing its random behaviour through a continuous probability distribution. A way to overcome this limitation, explored in Sections 3 and 4, consists of adapting to CPCDs the financial risk measures VaR and CVaR. Hull ([25], pp. 305–328) offered a synthetic approach to both measures and their main properties. Alexander [26] presented a comprehensive study of these indicators. In this line, the TR model extends the denomination of CPCDs to the probability–consequence combinations, based on VaR and CVaR, that it builds up through the half-normal distribution. Among other strong points, VaR and CVaR have the advantage of their links with the properties of coherent risk measures enunciated by Artzner et al. [15] (monotonicity, translation invariance, positive homogeneity, and subadditivity). As known, CVaR systematically fulfils the four properties, while VaR, in certain circumstances, fails subadditivity. However, VaR fulfils subadditivity for normally distributed returns ([26], p. 39; [27], p. 135) and in other more general cases as well [28]. The research developed in this paper relies on the interactions of the concepts of truncated probability distributions, risk matrices, probability–consequence diagrams, VaR, and CVaR. Risk matrices have their strong point in their communication capacity and the weakest point in their limitations for ranking medium-risk, i.e., yellow, scenarios correctly. CPCDs generate coherent risk rankings, but their analytical capacity depends on the risk measure they incorporate. In other words, a ranking may be correct for a specific risk measure, but this measure may be perfectible. Estimating the risk level through the product between probability and severity restrains the potential for exploring the main risk features of the studied scenarios. VaR and CVaR are risk measures with a higher analytical capacity. However, they have not been adapted to the comparison between probability and severity, which is the primary goal of risk matrices and CPCDs. The fact that the probability–severity approach only deals with negative scenarios makes truncated probability distributions an excellent tool for their study. Risk analysis, including VaR, has widely used these distributions. However, research in academic text databases (Google Scholar, ABI inform, Science Direct, and others) does not show evidence of their adaptation to the probability–severity analysis. To sum up, this synthesis of the risk analysis literature raises the following research question: How can the half-normal distribution improve risk analysis based on the interaction between probability and severity by adapting VaR and CVaR to CPCDs? The paper is structured as follows. Section 2 defines the Hazards Index-at-Risk (HIaR) and studies its main properties. This index constitutes an adaptation of the VaR to the CPCDs through the half-normal distribution. Section 3 builds HIaR diagrams, comparing them with risk matrices. Section 4 focuses on the study of the Expected Hazards Damage (EHD), which extends HiaR to the expected loss, again through the half-normal distribution. Section 5 compares the TR model with traditional risk matrices, centring the functional relationships between EHD and the expected loss as estimated in risk matrices. Section 6 studies the conditional HIaR (CHIaR), underlining the difference between active and latent risks that it helps to identify. Section 7 presents a subjective approach to the previous topics. The extension of the TR model to the generalised half-normal distribution, focusing on the interpretation and role of its shape parameter, is shown in Section 8. Section 9 discusses the results of this paper, and Section 10 summarises its conclusions. The two appendices present, respectively, a summary of the main properties of the half-normal distribution and obtainment of the expected hazards damage for the generalised half-normal distribution. Mathematica is the software used in the preparation of our paper. Mathematics 2020, 8, 2080 4 of 27 2. Defining the Hazards Index-at-Risk The TR model initiated in this section starts by adapting VaR to CPCDs without constraining its results to the financial consequences of risk. For this reason, monetary units are put aside systematically. The model centres on a generic asset (A) that may be damaged by the hazards (H) generated by the risk under2020, Mathematics study 8, x(R). FORThe PEERintegrity REVIEW of the asset concerning risk impacts, i.e., hazards, is represented 4 ofby 28 the asset’s safety index (SF). Namely, SFAR stands for the safety index of asset A in the face of risk R. in To the face avoid of risk R. working Tonegative with avoid working with units, the TRnegative units, thethe model expresses TRrisk model expresses impacts the risk in positive impacts points that, in positive next, points subtracts from that, thenext, subtracts current from value of thethe current safety indexvalue of theitssafety to obtain index to new value. Theobtain itsvalue initial new value. The initial of the safety index value of the safety is normalised index at 100 is normalised points. Hazardsat 100 points. reduce Hazards this value and reduce thisturn even may value and it into even may turn it into a negative one. In this case, the damage caused by hazards a negative one. In this case, the damage caused by hazards goes beyond their impacts on the asset. goes beyond their impacts Scheme 1ondepicts the asset. theseScheme 1 depicts these interactions. interactions. Scheme 1. Interacting assets and risk: this conceptual map summarises the interaction between assets Scheme 1. Interacting assets and risk: this conceptual map summarises the interaction between assets and risk expressed through the changes in the safety index caused by hazards. and risk expressed through the changes in the safety index caused by hazards. Let us recall that VaR is defined as the maximum loss that an institution faces in the next period Let us recall that VaR is defined as the maximum loss that an institution faces in the next period with a certain probability, p. This means, in other words, that the institution does not expect a higher with a certain probability, p. This means, in other words, that the institution does not expect a higher loss than the VaR with probability 1 − p (confidence level). For the case of banks, the usual choice loss than the VaR with probability 1 − p (confidence level). For the case of banks, the usual choice is is 1% probability (i.e., 99% confidence level) for ten days. VaR is also used in corporate risk with 1% probability (i.e., 99% confidence level) for ten days. VaR is also used in corporate risk with longer longer horizons and lower confidence levels. A normal distribution of gains is the most usual in VaR horizons and lower confidence levels. A normal distribution of gains is the most usual in VaR calculations, although other distributions are also applied. This distribution includes gains and losses calculations, although other distributions are also applied. This distribution includes gains and losses as negative gains ([25] p. 305). The TR model displays a setting where only negative impacts, namely as negative gains ([25] p. 305). The TR model displays a setting where only negative impacts, namely losses, are possible. These losses consist of reductions in the safety index. To incorporate them into losses, are possible. These losses consist of reductions in the safety index. To incorporate them into the TR model, we define the risk rate (x) as the relative change in the initial value of the safety index the TR model, produced we define by a hazard. Thethe risk risk rate rate ( x ) as the expresses systematically relative change in thein a reduction initial valueindex. the safety of the Losses safety are calculated index produced as by the aproduct hazard.ofThe the risk initial value rate of the safetyexpresses systematically index by the current value a reduction in theofsafety the risk rate. index. Losses arethe Formally, calculated as the loss at the product moment t (Lof t ) the andinitial value the value ofof thethe safety safety index index by the after loss (SFvalue thiscurrent t ) are of the risk rate. Formally, the loss at the moment t ( Lt ) and the value of the safety index after this loss Lt = SFt−1 xt (1) ( SFt ) are SFt = SFt−1 (1 − xt ) (2) Lt = SFt −1 xt (1) SFt = SFt −1 (1 − xt ) (2) The TR model assumes that the risk rate follows a half-normal probability distribution because it can take only positive signs under the assumptions previously introduced. Since losses are Mathematics 2020, 8, 2080 5 of 27 The TR model assumes that the risk rate follows a half-normal probability distribution because it can take only positive signs under the assumptions previously introduced. Since losses are expressed as the product between the constant initial value of the safety index and this rate, they follow a half-normal distribution as well. Appendix A summarises the main properties of the half-normal probability Mathematicsdistribution usedREVIEW 2020, 8, x FOR PEER in this paper. 5 of 29 We define HIaR as the maximum potential reduction that the safety index can suffer in the next period We withdefine HIaR as the a probability maximum equal to or lower potential thanreduction the maximum that the safety index reasonable can sufferof probability inthe the hazard next period with a probability equal to or lower than the maximum reasonable occurrence. HIaR shares with VaR the property of being quantiles. Alexander ([26], p. 13) introducedprobability of the hazard VaR occurrence. by writing HIaR “Valueshares withisVaR at risk the that a loss property we areof being fairly quantiles. sure that Alexander will not be([26], p. 13) if exceeded introduced the current VaR byiswriting portfolio held over“Value some at risk is a of period loss that we time”. are fairlythis Paralleling suredefinition, that will not be exceeded HIaR if the current can be conceived as the portfolio is held over some period of time”. Paralleling hazards impact that the analyst is reasonably sure will not be exceeded. this definition, HIaR can be conceived as the hazards impact that the analyst is reasonably sure will not be exceeded. The comparison between the areas of corresponding probability distributions at the left-hand side and Theat comparison the right-hand between thethe side of areas of corresponding quantiles (VaR and HIaR) probability distributions clarify the meanings, at the left-handthe especially side and at the right-hand side of the quantiles (VaR and HIaR) clarify the meanings, especially the meaning of the confidence levels associated with VaR and HIaR. The area under the probability density meaning of the confidence levels associated with VaR and HIaR. The area under the probability function on the left-hand side of VaR expresses the probability of losing more than VaR. The area on the density function on the left-hand side of VaR expresses the probability of losing more than VaR. The right-hand side of HIaR measures the probability of suffering damage greater than HIaR. Therefore, area on the right-hand side of HIaR measures the probability of suffering damage greater than HIaR. the corresponding confidence levels are referred to as opposite areas. As known, the confidence level Therefore, the corresponding confidence levels are referred to as opposite areas. As known, the forconfidence VaR is measured level forbyVaRtheisarea on its right-hand measured by the area onside. its right-hand side. Conversely, Conversely, thethe confidence confidencelevellevel for HIaR is measured for HIaR by the area is measured on its by the left-hand area side. Associating on its left-hand side. HIaR with the probability expected by the analyst, the area on its left-hand Associating HIaR with the probability expected by the analyst, the area on its left-hand side side expresses theexpresses maximum expected probability the maximum of the expected occurrence probability ofoccurrence of the the hazards. In hazards. of the contrast,Inthe area on contrast, theits right-hand area on its right- side expresses hand sidethe probability expresses of experiencing the probability damage damage of experiencing greater thangreaterHIaR, than i.e., HIaR,it i.e., can itbecan regarded be regardedas the surprise probability. as the surprise For thisFor probability. reason, the probability this reason, of having the probability damage of having equalequal damage to ortolower thanthan or lower HIaR is aHIaR confidence level. Figure is a confidence 1 represents level. the comparison Figure 1 represents between VaR the comparison and HIaR. between VaR The and probability HIaR. The of probability facing damageofgreater facingthan damageHIaR, greater whichthan HIaR,the generates which generates CHIaR, the CHIaR, is the centre of theisanalysis the centre of the in presented analysis Section 6. presented in Section 6. Figure 1. Value-at-Risk (VaR) vs. Hazards Index-at-Risk (HIaR): VaR is estimated for a random rate of Figure 1. Value-at-Risk (VaR) vs. Hazards Index-at-Risk (HIaR): VaR is estimated for a random rate return that follows a normal distribution with mean 0 and standard deviation 0.20. The confidence level of return that follows a normal distribution with mean 0 and standard deviation 0.20. The confidence is set at 0.90. HIaR is estimated for a half-normal distribution. The standard deviation for its equivalent level is set at 0.90. HIaR is estimated for a half-normal distribution. The standard deviation for its normal distribution equivalent normal is set at 0.20,isand distribution set the confidence at 0.20, and thelevel is 0.90 level confidence as well. Theasdashed is 0.90 vertical well. The lines dashed point out VaR vertical linesand HIaR point on the out VaR andright HIaR andon left, respectively. the right The confidence and left, respectively. Thelevel for VaR confidence is the level forarea VaRon theisright-hand side of the dashed line labelled as VaR, while for HIaR, the confidence the area on the right-hand side of the dashed line labelled as VaR, while for HIaR, the confidence level is the area onlevel the left-hand side of the dashed line labelled as HIaR. is the area on the left-hand side of the dashed line labelled as HIaR. Calculation of HIaR requires estimation of the standard deviation of the risk rate and deciding the confidence level. The standard deviation can indistinctively be referred to as the half-normal distribution or its equivalent value for the normal distribution due to the univocal relationship that exists between both (see Appendix A, Equation (A5)). The formulas that follow incorporate the standard deviation of the equivalent normal distribution. As known, the half-normal probability distribution function is as follows: x2 1 2 − f ( x) = e 2σ 2 dx (3) σ π Mathematics 2020, 8, 2080 6 of 27 Calculation of HIaR requires estimation of the standard deviation of the risk rate and deciding the confidence level. The standard deviation can indistinctively be referred to as the half-normal distribution or its equivalent value for the normal distribution due to the univocal relationship that exists between both (see Appendix A, Equation (A5)). The formulas that follow incorporate the standard deviation of the equivalent normal distribution. As known, the half-normal probability distribution function is as follows: r 1 2 − x22 f (x) = e 2σ dx (3) σ π Designating by γσ,c f the value of risk rate x that matches the standard deviation σ and the confidence level c f , we can write the following: γσ,c f r x2 Z 1 2 − e 2σ2 dx = c f (4) σ π 0 Calculating this integral, we obtain γσ,c f " # Er f √ = cf (5) 2σ and solve it for γ: √ γσ,c f = σ 2Er f −1 [c f ] (6) Er f −1 [c f ] stands for the inverse of the error function of the confidence level, i.e., the inverse of the following: Zc f 2 2 Er f [c f ] = √ e−x dx (7) π 0 which is calculated through numerical methods. At this point, HIaR is obtained straightforwardly by multiplying γσ,c f by the initial value of the safety index: HIaRσ,c f = SF · γσ,c f (8) Since the notation γσ,c f stands for the risk rate that, once multiplied by the initial value of the safety index, generates HIaR, it can be the denominated HIaR rate. In that the former refers to expected values while the latter expresses actual values. Comparing their dependent variables, L stands for an actual loss whereas HIaR stands for an expected loss. It stems from Equation (6) that the expected value of the HIaR rate is proportional to the standard deviation. Thus, designating by γ1,c f the expected HIaR rate for standard deviation 1 and confidence level c f , we have the following: γσ,c f = σ · γ1,c f (9) where √ γ1,c f = 2 · Er f −1 [c f ] (10) which is the value of γ for σ = 1 and the corresponding c f . Thus, HIaR can be expressed as follows: HIaR = SF · σ · γ1,c f (11) Switching to the logarithmic scale in Equations (9) and (11), we can immediately separate the effects of the standard deviation and the confidence level. Mathematics 2020, 8, 2080 7 of 27 Mathematics 2020, 8, x FOR PEER REVIEW 7 of 28 (b) TheThe probability effect, expression of ⋅Erfshown the HIaR2 rate −1 [ ] cf in, that expresses (6) conveys thatthe theinfluence HIaR rateofturns the out confidence to be thelevel on product the HIaR rate because it can be interpreted as the normalised HIaR rate for σ = 1, i.e., γ of two effects: . 1,cf (a) The severity effect σ that associates the HIaR rate to the severity of the hazards impact. These effects parallel the√traditional approach of risk matrices, where the impact size measures (b) severity the The probability effect andeffect, 2·Er f −1occurrence the estimated [c f ], that expresses thedirectly probability influence of the confidence expresses level effect. the probability on the HIaR rate because it can be interpreted as the normalised HIaR In the TR model, both effects are filtered through the HIaR formula shown in Equationrate for σ = 1, i.e., γ f . 1,c(6). Their productThese effects parallel the traditional approach of risk matrices, where the impact size measuresthat with the safety index in (11) simply means a change of scale. It stems from Equation (6) the the HIaR rate is proportional to the severity effect. Centring on the probability effect, we severity effect and the estimated occurrence probability directly expresses the probability effect. In the observe that itTR is model, a monotonously increasing both effects function are filtered through cf HIaR of the that approaches zero when formula shown cf approaches in Equation zero as (6). Their product with It well. the equates whenincf safety 1index (11)equates Erf 1 /a change simply means 2  (0.6827 of scale. It stems from approximately) andEquation approaches(6) positive that the HIaR rate is proportional to the severity effect. Centring on the probability effect, we observe that it infinity when cf approaches 1. Thus, the probability effect is lower than 1 for cf between zero is a monotonously increasing functionh √ i of c f that approaches zero when c f approaches zero as well. and Erf 11when It equates / 2 c f equates Er f 1/than and greater 2 (0.6827 1 for approximately) Erf 1positive and approaches probabilities between / 2  infinity and 1. when h √ i The c f approaches consequence of1.these Thus,properties the probability effect for the HIaR is loweristhan 1 for c f between zero and Er f 1/ 2 and h rate √ i that the probability effect reduces HiaR for probabilities lower than Erf 1 / 2  and increases this rate for probabilities higher greater than 1 for probabilities between Er f 1/ 2 and 1. The consequence of these properties for the HIaR rate is that the probability effect reduces HiaR for probabilities lower than Er f h1/ √2ithan and Erf 1 / 2  . h √ i   increases this rate for probabilities higher than Er f 1/ 2 . Figure 22 shows Figure shows thethe evolution evolution of of HIaR HIaR asas an an increasing increasing function function of of the the standard standard deviation deviation and and the confidence the confidence level. level. The Thecolours coloursare arescaled scaledaccording accordingto tothe the traditional traditional risk risk matrices matrices convention: convention: green for low risk, yellow for medium risk, and red for high risk, with added green for low risk, yellow for medium risk, and red for high risk, with added purple for very purple for very high high risk. The right-hand-side plot includes a plane, denoted by grey lines, set up at 100 risk. The right-hand-side plot includes a plane, denoted by grey lines, set up at 100 HIaR points to HIaR points to visualise the visualise the probability–severity probability–severity combinations combinationsthatthatexpress express damage damage greater greater than than the the initial initial value value of of the safety index. the safety index. 2. HIaR Figure 2. Figure HIaRfunction: thethe function: left-hand-side plot represents left-hand-side HIaR as plot represents a function HIaR as a of probability function (confidence of probability level) and severity (standard deviation). The right-hand-side plot delimits through the added (confidence level) and severity (standard deviation). The right-hand-side plot delimits through planethe the HIaR value for which damages go beyond the initial value of the safety index (100 points). added plane the HIaR value for which damages go beyond the initial value of the safety index (100 points). 3. HIaR Diagrams vs. Risk Matrices TheDiagrams 3. HIaR conceptual centre vs. Riskof risk matrices consists of estimating the risk levels of the potential hazards Matrices under analysis by interacting the probability and severity of their impacts. Often, the numerical values The conceptual centre of risk matrices consists of estimating the risk levels of the potential of the probability and the impact level are outcomes of informed but subjective decision-makers’ criteria. hazards under analysis by interacting the probability and severity of their impacts. Often, the The most common estimation of risk levels consists of the product between probability and severity. numerical values of the probability and the impact level are outcomes of informed but subjective Next, the expected impacts are gathered into homogeneous categories that receive a similar treatment decision-makers’ criteria. The most common estimation of risk levels consists of the product between in the organization’s risk policy. The different risk categories are determined according to the values probability and severity. Next, the expected impacts are gathered into homogeneous categories that of probabilities and impacts that managers regard as appropriate for becoming boundaries between receive a similar treatment in the organization’s risk policy. The different risk categories are determined according to the values of probabilities and impacts that managers regard as appropriate Mathematics 2020, 8, 2080 8 of 27 the risk categories. The horizontal and vertical boundaries generate the characteristic grid shape of risk matrices. Despite their intuitive appeal, risk matrices cannot achieve a complete homogeneous classification of risk levels as pointed out in the Introduction section. CPCDs are also based on the interaction between probability and severity. Frequently, CPCDs estimate risk levels as the product between both parameters. In this case, the difference between CPCDs and risk matrices consists of the different structures of their diagrams, although they share the same method for estimating risk levels. The HIaR diagrams built up in this section constitute a variant of CPCDs. Equal to CPCDs, HIaR diagrams divide risk zones through continuous iso-risk lines, now iso-HIaR lines. HIaR diagrams incorporate not only the intuition but also the rigour of VaR into CPCD analysis. A central feature of risk matrices is their division in action zones denoted by different risk categories. In this respect, a risk category is constituted by the values of risk levels that deserve a similar course of action. Any risk matrix is divided at least into three action zones. Each zone recommends a different kind of action. For instance, “wait-and-see”, “short-term-management”, and “structural change”. The TR model replaces the rectangular delimitation of action zones used in risk matrices by introducing iso-HIaR lines as the boundaries of action zones. An iso-HIAR line consists of the combination of confidence levels and standard deviations that generate the same value of HIAR. Designating by HIaR j the constant value of HIaR that determines the iso-HIaR line j, by substituting in (11) HIaR by HIaR j and γ1,c f for its expression according to (10), we obtain the function that determines the pairs of σ and c f that generate the iso-HIaR line for HIaR j . Any HIaR diagram needs defining  boundary  Hia R levels, which become the boundaries that delimit the action zones. The HIaR rate γσ,c f shown in (6) is proven to be a useful tool for deciding the boundaries of the HIaR diagram because it can be interpreted as the percentage that HIaR represents on the safety index, namely the percentage of the safety index that HIaR would destroy in the case of being real. Then, HIaR boundaries can be decided according to the HIaR rates that justify a change in the action zone, according to the analysts’ criteria. HIaR matrices may complement HIaR diagrams. A HIaR matrix consists of displaying in matrix form the HIaR values that match the chosen combinations of standard deviations, shown in the upper row, and the confidence levels shown in the left-hand-side column. Due to this discrete structure, a HIaR matrix is not appropriate for separating iso-risk zones. Table 1 shows the HIaR matrix for the given standard deviations and confidence levels. These HIaR values can be interpreted as percentages. For instance, for a standard deviation of 0.25 (severity) and a confidence level (probability) of 0.25, the expected damage in the safety index is 7.97%. However, if severity equates 0.75 and probability equates 0.95, then the expected damage goes beyond the current value of the safety index (100 points) reaching 147% of it, which means that the damage exceeds 47% of the value embedded in the safety index. At the same time, the first column shows the probability effect, i.e., γ1 . Table 1. HIaR matrix. σ 0.01 0.05 0.25 0.5 0.75 1 1.5 cf 0.99 2.58 12.88 64.40 128.80 193.20 257.60 386.40 0.95 1.96 9.80 49.00 98.00 147.00 196.00 294.00 0.75 1.15 5.75 28.76 57.52 86.28 115.00 172.60 0.5 0.67 3.37 16.86 33.72 50.59 67.45 101.20 0.25 0.32 1.59 7.97 15.93 23.90 31.86 47.80 0.05 0.06 0.31 1.57 3.14 4.70 6.27 9.41 0.01 0.01 0.06 0.31 0.63 0.94 1.25 1.88 Table 1 shows the values of HIaR for the given values of the standard deviation and confidence levels. Mathematics 2020, 8, 2080 9 of 27 Mathematics 2020, 8, x FOR PEER REVIEW 9 of 28 Figure 3 shows the HIaR diagram as a two-dimensional plot and three three-dimensional plots. Figure Their data 3 shows are the samethe HIaRgenerated as that diagram asbyathe two-dimensional HIaR matrix inplot and Table 1. three three-dimensional As Figures plots. 2 and 3 include four Their data are the same as that generated by the HIaR matrix in Table 1. As Figures 2 and 3 include risk levels denoted by green (low risk), yellow (medium risk), red (high risk), and purple (extreme risk) four risk levels denoted by green (low risk), yellow (medium risk), red (high risk), and purple colours. The boundary levels are set up at 25 points for starting the yellow level, 75 for starting the red (extreme risk) colours. The boundary levels are set up at 25 points for starting the yellow level, 75 for level, and 150 for starting the purple level. The three three-dimensional plots display the figure from starting the red level, and 150 for starting the purple level. The three three-dimensional plots display three different points of view on behalf of clarity. In them, iso-HiaR planes have substituted iso-HiaR the figure from three different points of view on behalf of clarity. In them, iso-HiaR planes have lines. Throughout the paper, in the two-dimensional diagrams, the abscissa axis denotes the standard substituted iso-HiaR lines. Throughout the paper, in the two-dimensional diagrams, the abscissa axis deviation and the ordinate axis denotes the confidence level. denotes the standard deviation and the ordinate axis denotes the confidence level. Figure Figure 3. 3.HIaR HIaRdiagram: diagram: this this figure figure displays displays the the HIaR HIaR diagrams diagramsthat thatcorrespond correspondtotothe thematrix matrix presented presented in Table in Table 1. In1.theInthree-dimensional the three-dimensional diagrams, diagrams, theare the axes axes the are the standard standard deviationdeviation (severity), the(severity), confidence thelevel confidence level (probability), (probability), and HIaR. Boundaries and HIaR. Boundaries are placed are placed at 25, at 25, 75, and 75, and 150. 150. 4. The Expected 4. The Hazards Expected HazardsDamage Damage HIaR would HIaR would become becomeincomplete incompletewithout without connecting connecting itit to to the theexpected expectedloss lossinincase caseofofa ahazard hazard taking place. We define the Expected Hazards Damage (EHD) as the expected loss taking place. We define the Expected Hazards Damage (EHD) as the expected loss due to the risk under due to the risk under i.e., analysis, analysis, i.e., the loss the expected expected if theloss if the hazard hazard real. becomes becomes real. Recalling Recalling that HIaRthat HIaR expresses expresses maximum lossmaximum loss index in the safety in the (SF) safety indexthe under (SF) under theprobability estimated c f , the EHDcfcan estimated probability , the be EHD writtencanasbefollows: written as follows: γσ,c f r γ σ ,cf 1 x2x2 Z 1 EHD = 1 SF 1x 22e−−2σ2σ2 2dx  (12) EHD = c f SF σ x π e dx (12) cf 0 0 σ π where 1/c1f cf where adjusts thethe adjusts probability of of probability the half-normal the half-normaldistribution distribution to to the area areabetween betweenzero andcf zeroand c f.. Solving the integral in Equation (12), we obtain the following: Mathematics 2020, 8, x2080 FOR PEER REVIEW 27 10 of 28 Mathematics 2020, 8, x FOR PEER REVIEW 10 of 28 (( )) Solving the integral in Equation (12), we1obtain the − Erf−1following: −1 [cf2 ]2 2 EHD SF 1 11−−ee− Erf [cf ] EHD == SF 2 σ σ (13)(13) cf cf  ππ r 1 −1 2 2 EHD = SF 1 − e−Er f [c f ] σ (13) Paralleling Parallelingthe thebreaking breaking down down ofof the the HIaR HIaR betweenthe c f between π theprobability probability effect effect andand thethe severity severity effect, effect, this thisequation equationextends extendsthis thisdecomposition decomposition to toEHD. EHD.Now,Now, σσ continues continues being being thethe severity severity effect, effect, while while Paralleling the breaking down of the HIaR between the probability effect and the severity effect, (( 1 1 equation )) 22this decomposition to EHD. Now, σ continues being the severity effect, while 2 − Erf −1−[1cf[ cf] ]2 this  11 −−ee− Erf extends becomes becomes the probability probabilityeffect. effect. cf 1 cf 2π πbecomes the probability effect. q 2 1 − e−Er f −1 [c f ] cf π ItIt stemsfrom from Equation (12) (12) EHD consists of the mathematical expectation of thethe probability It stems stems from Equation Equation (12) that that EHD EHD consists consists of of the the mathematical mathematical expectation expectation of of the probability probability distributionofofthe distribution thehazards hazards impacts impacts that do not exceed the maximum expected impact, i.e., do donot distribution of the hazards impacts that do not exceed the maximum expected impact, i.e., that do not exceed the maximum expected impact, i.e., do notnot exceed exceed HIaR. HIaR. Thus, Thus, this this distribution distribution is is a a half-normal half-normal truncated truncated at the at theHIaR HIaRpoint. Equation point. Equation(12) shows (12) shows exceed HIaR. Thus, this distribution is a half-normal truncated at the HIaR point. Equation (12) shows thisproperty this propertyby bytruncating truncating the the half-normal half-normal distribution distribution ofofthe the risk rate x xat at thethe HIaR rate γ σ ,cfγ . . this property by truncating the half-normal distribution of the riskrisk raterate x at the HIaR HIaR rate γσ,crate f . Figure σ ,cf 4 Figure illustrates Figure 4 illustrates this property. 4 illustrates this property. The The left-hand this property. left-hand side side ofside The left-hand of Figure Figure 5 shows of Figure 5 shows the the EHD 5 shows EHD plot expressing plot expressing the EHD plot expressing EHD EHDEHD as a asa afunction function of σ of of σσ and cf .. Its Its right-hand right-hand side shows the EHDEHDdiagrams. function as and c f .and cf Its right-hand side shows sidethe EHD shows diagrams. the diagrams. Figure 4. Expected hazards damage vs. total potential damage: Figure 4 compares the half-normal Figure 4. Expected Expected distribution hazards of the risk ( ) x truncated rate damage vs. total total potential potential at the damage: HIaR rate ( γσ ,cf Figure Figure )4 compares the half-normal  with4the complete half-normal distribution of distribution of the riskrate distribution ofthex risk . The values ( ) rate (xx) truncated of truncated the standard at the at theHIaR deviationrate ( HIaR rate γγσ,c f with and theσ confidence ,cf ) withthe the complete complete level half-normal half-normal are, respectively, distribution of x. The values of the standard deviation and the confidence level are, respectively, 0.25 0.25 and distribution0.7. and 0.7. The of The x . The values corresponding corresponding HIaR HIaR of rate rate is the standard 0.2591. is 0.2591. deviation and the confidence level are, respectively, 0.25 and 0.7. The corresponding HIaR rate is 0.2591. Figure Expected Figure 5.5. Expected Hazards Hazards Damage Damage (EHD)(EHD) function function and diagrams: and diagrams: the side the left-hand left-hand side of Figure 5 of Figure 5 displays displays the three-dimensional the three-dimensional EHD plot EHD plot to according according to the the function function shown shown(13). in Equation in Equation The right-(13). The right-hand side of this figurethe displays the EHD diagram for theassame data diagram as the HIaR diagram Figure 5. Expected Hazards Damage (EHD) function and diagrams: the left-hand sideshown hand side of this figure displays EHD diagram for the same data the HIaR in 5 of Figure shown Figurein3.Figure The 3. The boundary boundary levels havelevels been have placedbeen at placed 10, 50, at 90. and 10, 50, and 90. displays the three-dimensional EHD plot according to the function shown in Equation (13). The right- hand side of this figure displays the EHD diagram for the same data as the HIaR diagram shown in Figure 3. The boundary levels have been placed at 10, 50, and 90. Mathematics 2020, 8, x FOR PEER REVIEW 11 of 28 Mathematics 2020, 8, 2080 11 of 27 EHD can be expressed as a function of HIaR, which clarifies the relationship between both. Solving (11) for σ and substituting γ 1,cf according to (10), we obtain EHD can be expressed as a function of HIaR, which clarifies the relationship between both. Solving (11) for σ and substituting γ1,c f according HIaR to (10), we obtain σ= ⋅ (14) SF 2 ⋅ Erf −1 cf HIaR [ ] σ= √ · (14) and substituting (14) into (13), we obtain SF 2 · Er f −1 [c f ] (1 − e ) 2 and substituting (14) into (13), we obtain − Erf −1 [ cf ] EHD =  HIaR (15) f −1−[c1 f[] cf ]  π1 −cfe−Er ⋅ Erf 2 EHD = √ HIaR (15) πc f and which shows that the relationship between EHD −1 [c f ] · Er f HIaR is ruled by the function of the HIaR–EHD connector which shows(ϑthat ) : the relationship between EHD and HIaR is ruled by the function of the HIaR–EHD (1 − e ) connector (ϑ): 2  −1  − Erf −1 [cf ]2 1 − e−Er f [c f ] ϑϑ = = √ (16) ππccff ·⋅ Er Erff −1−1[c fcf] [ ] (16) that that has has the the confidence confidencelevel levelas asthe the unique unique independent independentvariable. variable. Figure Figure 66 illustrates illustrates the the relationship relationship between betweenEHD EHDandandHIaR. HIaR. EHD-HIaR Driver (ϑ ) EHD as a Function of cf and HIaR Figure 6. EHD vs. HiaR: the left-hand side of Figure 6 depicts the evolution of the HIaR–EHD connector Figure 6. EHD vs. HiaR: the left-hand side of Figure 6 depicts the evolution of the HIaR–EHD according to the confidence level, as shown by Equation (16). Its right-hand side shows the evolution connector according to the confidence level, as shown by Equation (16). Its right-hand side shows the of the EHD according to the HIaR and the confidence level. evolution of the EHD according to the HIaR and the confidence level. 5. The Expected Hazards Damage vs. Traditional Risk Matrices Approach 5. The Expected Hazards Damage vs. Traditional Risk Matrices Approach Analysing the traditional risk matrices approach from EHD reveals hidden information of the Analysing hazard the traditional under analysis. risk matrices The traditional approach approach obtainsfrom theEHD reveals expected hidden impact information through of the the product hazard under analysis. The traditional approach obtains the expected impact through the product between the estimated probability and the size of the impact. Henceforth, we will call this product between the Expected estimated Traditional probability Impact (ETI): and the size of the impact. Henceforth, we will call this product Expected Traditional Impact (ETI): ETI = p · IS (17) where p stands for probability and IS stands ETI = p ⋅ IS for impact size. (17) whereBoth ETI and p stands EHD for share being probability andexpected IS standsvalues. The difference for impact size. between them lies in the obtainment processes. Both ETI and EHD share being expected values. The differenceprobability The calculation of EHD substitutes the direct product between between themand impact lies insize the shown in (17) by expression (13), where the expected value of the half-normal distribution obtainment processes. The calculation of EHD substitutes the direct product between probability and between zero andsize impact theshown HIaR rate is adjusted in (17) according by expression (13),towhere the confidence the expectedlevel. Theofrelationship value between the half-normal ETI and distribution EHD can be explored analytically. The algorithm we present next transforms ETI between zero and the HIaR rate is adjusted according to the confidence level. The relationship in EHD, revealing hidden information in the former. The algorithm proceeds as follows: Mathematics 2020, 8, x FOR PEER REVIEW 12 of 28 between ETI and EHD can be explored analytically. The algorithm we present next transforms ETI in EHD, revealing hidden information in the former. The algorithm proceeds as follows: Mathematics 2020, 8, 2080 12 of 27 (1) The probability and the impact size are accepted as data to obtain ETI according to (17). (2) The probability is assimilated to the confidence level ( p = cf ) . (1) The probability and the impact size are accepted as data to obtain ETI according to (17). (3) The impact size is assimilated to HIAR ( IS = HIaR (2) The probability is assimilated to the confidence level (p = c f ). ) . (4) From (3) the confidence The impact level and to size is assimilated HIaR, HIAR we(IS obtain the)standard = HIaR . deviation embedded in the impact size (IS) through Equation (14) by substituting HIaR (4) From the confidence level and HIaR, we obtain the standard deviation for IS. The knowledge embedded on this in the standard impact size deviation (IS) through is not necessary Equation (14) for relating ETI HIaR by substituting with EHD, for IS.but Theitknowledge can be compared on this with historical standard and deviation estimated volatilities is not necessary to evaluate for relating ETI with theEHD, soundness of the but it can ETI calculated be compared through the with historical andtraditional estimated method. volatilities to evaluate the soundness of the ETI calculated through the traditional method. (5) Having (5) Having assimilated assimilated IS IS toto HIaR, HIaR, itit stems stems from from (17) (17) that that HIaRHIaR can can be be equated equated to the ratio to the between ratio between ETI and cf because cf substitutes p in the ETI and cf because cf substitutes p in the TR model. TR model. (6) Substituting HIaR (6) Substituting HIaR inin (15) (15) by by the the quotient quotient between between ETI ETI and and cf, cf, we we obtain obtain the the relationship relationship between between ETI and EHD: ETI and EHD:   ( ) e−Er 1 −− Erf f −1 [c f ]2 −1 [cf ]2 EHD =1 − √e 2 ETI (18) (18) EHD = πc f Er f −1 [c fET ] I π cf 2 Erf −1 [ cf ] Thus, the function that links EHD with ETI is as follows: Thus, the function that links EHD with ETI is as follows: ( )   −1 22 11− e − Erff −1 [[ccff ]] −Er −e φ= √ (19) φ = πc f22 Er f −1−1[c f ] (19) π cf Erf [cf ] Figure 7 illustrates the relationship between EHD and ETI. Figure 7 illustrates the relationship between EHD and ETI. Figure 7.7. EHD Figure EHDvs.vs.Expected Traditional Expected Impact Traditional (ETI):(ETI): Impact the left-hand side of side the left-hand Figure of 7Figure shows the evolution 7 shows the the EHD–ETI connector (φ) according to the confidence level. Function φ evolution of the EHD–ETI connector ( φ ) according to the confidence level. Function φ of monotonously decreases with the confidence level, equating 1 for a confidence level equal to 0.482779. Thus, for this value of monotonously decreases with the confidence level, equating 1 for a confidence level equal to 0.482779. confidence level, estimating ETI turns out to be equivalent to estimating EHD. The right-hand side Thus, for this value of confidence level, estimating ETI turns out to be equivalent to estimating EHD. displays the evolution of EHD according to ETI and the confidence level. The right-hand side displays the evolution of EHD according to ETI and the confidence level. 6. On Active and Latent Risks: Conditional HIaR 6. On Active and Latent Risks: Conditional HIaR This section delves into the comparative analysis between the maximum expected probability Thissurprise and the section probability delves into introduced the comparative analysis in Section between 2 when the maximum considering damage expected greater probability than HIaR. and evolution The the surprise fromprobability the HIAR introduced in Section to the analysis 2 when values of the expected considering damage associated withgreater thanprecise it requires HIaR. The evolution from the HIAR to the analysis of the expected values associated with it requires interpretation of the probability that determines the HIaR, henceforth, the HIaR probability. The binary precise interpretation approach of the probability of traditional thatincorporates risk matrices determines the twoHIaR, henceforth, probabilities the HIaR exclusively: theprobability. The one associated binary with theapproach occurrence of of traditional the hazardriskandmatrices incorporates the one that twoidea discards the probabilities exclusively: that the hazard will taketheplace. one associated with the occurrence of the hazard and the one that discards the idea that the In contrast, this paper centres its analysis on a continuous distribution, the half-normal. Obtaining hazard will take place. HIaR In the through contrast, this paper half-normal centres reveals distribution its analysis on a continuous the existence distribution, of three relevant levelstheof half-normal. the expected value: the total potential risk, the expected active risk, and the latent risk. Mathematics 2020, 8, 2080 13 of 27 The total potential risk consists of the expected value that stems from the whole probability distribution. Thus, it consists of the mathematical expectation of the potential hazard that the risk under analysis may generate. However, not necessarily, the entire potential risk constitutes an active threat. One part of it may be active, and the rest may be latent. The central feature of the latent part of the risk is that, under the current circumstances, it does not show any sign that it will produce any hazard on the horizon of the analysis. The expected active risk consists of the expected value of hazards that goes from zero to the HIaR probability, i.e., from a null probability to its maximum expected value. Finally, the latent risk is the expected value of the latent part of risk placed on the right-hand side of the HIaR probability. This latent risk is congruent with the risk associated with CVaR and placed on the left-hand side of VaR. The previous analysis has focused on the area between the origin and the confidence level of the half-normal distribution. The whole area of distribution and the area on the right-hand side of the confidence level complement its information. The total area under distribution can be interpreted as the total mathematical expectation (TME) of a hazard with volatility of the case under analysis. For the half-normal distribution, the unique variable of this area is the standard deviation: Z∞ r 1 2 − x22 TME = SF0 x e 2σ dx (20) σ π 0 Solving the integral in Equation (20), we obtain the following: r 2 TME = SF0 σ (21) π √ Since 2/π is equal to 0.7979, the TME approximately equates to 80% of the standard deviation multiplied by the initial value of the index. The right-hand-side area expresses the discarded impact, more specifically, the impact that analysts estimate that will not take place and, for this reason, discard it from the study. Paralleling EHD, the expected value on the right-hand side of the distribution leads to the Conditional HiaR (CHIaR), which means the potential impact excluded from the analysis: Z∞ r 1 1 2 − x22 CHIaR = SF0 x e 2σ dx (22) 1−cf σ π γσ,c f Solving the integral in Equation (22), we obtain the following: r 1 −1 2 2 CHIaR = SF0 e−Er f [c f ] σ (23) 1−cf π CHIaR can be regarded as the source of unexpected hazards known as black swans. CHIaR has a parallel meaning with CVaR because the former focuses on damage when the HIaR barrier is crossed and the latter focuses on losses when the VaR barrier is crossed. TME, EHD, and CHIaR fulfil the following properties: when the confidence level approaches 1, EHD approaches TME and CHIaR approaches zero. TME, in turn, is the weighted average of EHD and CHIaR with c f and (1 − c f ) as weighting coefficients. lim EHD = TME (24) c f →1 lim CHIaR = 0 (25) c f →1 Mathematics 2020, 8, 2080 14 of 27 Mathematics 2020, 8, x FOR PEER REVIEW 14 of 28 c f · EHD + (1 − c f ) · CHIaR = TME (26) cf ⋅ EHD + (1 − cf ) ⋅ CHIaR = TME (26) Figure 8 displays the probability distributions of total risk, active risk, and latent risk. Figure 8 displays the probability distributions of total risk, active risk, and latent risk. Total Risk Active Risk Latent Risk Active vs. Total Latent vs. Total Active vs. Latent Conjoint Plot Figure Figure 8. 8.Active Activevs. vs.latent latent and and total total risk: this this figure figurecompares comparesthe theprobability probability distributions distributions of of total total risk, active risk, active risk, risk,and andlatent latentrisk, risk,showing showing them in independent independentand andcombined combinedplots. plots. The The values values of the of the standard standard deviation deviation and andthethe confidence confidence level levelare, respectively, are, respectively,0.25 and 0.25 0.7. and The 0.7. Thecorresponding correspondingHIaR HIaRrate is rate 0.2591. It can be is 0.2591. It read can together with Figure be read together with4. Figure The dashed 4. Theline depicts dashed linethedepicts half-normal distribution the half-normal distribution before before its its truncation. Thetruncation. blue arrow The blue the shows arrow HIaR shows the point at HIaR whichpoint at which the the truncation truncation takes takes place. The pink place. zone The pink to corresponds zone EHD,corresponds to EHD, and the green zoneand the green to corresponds zone corresponds Conditional HIaRto (CHIaR). Conditional HIaR (CHIaR). 7. A Subjective Approach to Risk Analysis through the Half-Normal Distribution 7. A Subjective Approach to Risk Analysis through the Half-Normal Distribution The relevant contribution of subjective probabilities to decision-making reasoning is widely acknowledged. Their The relevant use in risk of contribution analysis is part subjective of this contribution probabilities as well. Karni to decision-making [29] systematised reasoning is widely acknowledged. their Their use inAndersen axiomatic foundations. risk analysis is [30] et al. part studied of this contribution as well. Karni the joint estimation [29] systematised of subjective probabilities andtheir riskaxiomatic attitudes.foundations. Wintle et al.Andersen [31] dealtetwith al. [30] studied thetranslation the numerical joint estimation of probabilities. of verbal subjective probabilities and risk attitudes. Wintle et al. [31] dealt with the numerical Aven and Reiners [32] presented an in-depth analysis of the interpretation of probabilities translation of verbal in risk probabilities. analysis Aven and with particular Reinerson[32] emphasis the presented meaning and an use in-depth analysisprobabilities. of subjective of the interpretation Goerlandtofand probabilities Reiners in risksubjective [33] applied analysis probabilities with particular emphasis on the meaningdiagrams. to probability–consequence and use Flage of subjective et al. [34] probabilities. Goerlandt and Reiners [33] applied subjective probabilities to probability–consequence developed a critical perspective on the use of subjective probabilities in risk analysis, showing how Mathematics 2020, 8, 2080 15 of 27 incorporating probability bounds may improve the performance of the analysis. Langdalen et al. [35] dealt with the identification of hidden assumptions in risk studies focusing on the subjective nature of risk assessment. This section aims to show how the analytical framework created by the TR model can be used also as a tool for subjective reasoning in risk analysis. In this way, generic risk perceptions can be turned into a set of linked variables, the coherence of which can be discussed through the lenses of the logic relationships embedded in the model. In concrete, this section presents an algorithm that, departing from subjective assumptions of the maximum downwards associated with the total risk and the active risk, delivers the values of the standard deviation, the confidence level, and the HIaR embedded in the assumed downwards. The algorithm proceeds as follows: (1) The hypothesis for the maximum downward coefficient of total risk leads to obtainment of the standard deviation by applying a property of binomial trees. In effect, binomial trees used in option pricing (Cox, Ross, and Rubinstein [36]) relate in approximate terms the upward and downward coefficients with the standard deviation. For the downward coefficient (d), the relationship for one-time period is as follows: e−σ = d (27) Based on this relationship, the analyst may obtain the standard deviation after estimating the maximum downward that s/he may expect in the safety index. This estimation includes the active and the latent risk, i.e., it refers to the total risk. (2) By introducing a hypothesis for the maximum downward coefficient of the active risk (d0 ), the analyst identifies the subjective HIaR embedded in this downward coefficient. Since HIaR represents the maximum expected reduction in the safety index, the relationship between d0 , SF, and HIaR is SF · (1 − d0 ) = HIaR (28) (3) To unveil the confidence level that stems from the obtained values of HIaR and the standard   deviation, we calculate, first, the value of the HIaR rate γ1,c f from Equation (11). Next, substituting it into Equation (10), we obtain the confidence level. Summarising, through subjective estimations of the downward coefficients for the total risk and the active risk (d and d0 ), the analyst obtains the set σ, HIaR, c f and starts revising its coherence from  his/her criteria and the information available for similar settings. The aim of this algorithm is not to substitute an in-depth analysis of subjective reasoning but to provide a first approach to turning subjective perceptions into a quantitative setting in a straightforward manner grounded on analytical tools. Scheme 2 presents a conceptual map of this algorithm. Mathematics 2020, 8, 2080 16 of 27 Mathematics 2020, 8, x FOR PEER REVIEW 16 of 28 Scheme 2. The Scheme subjective 2. The approach: subjective Scheme approach: 2 summarises Scheme in a conceptual 2 summarises map the in a conceptual subjective map approach the subjective to approach HIaR analysis presented to HIaR analysisin Section 7. presented in Section 7. 8. Exploring the Generalised Half-Normal Distribution 8. Exploring the Generalised Half-Normal Distribution Many probability distributions have been applied to risk analysis. Although this paper has Many probability distributions have been applied to risk analysis. Although this paper has centred on the half-normal distribution, the essence of its methodological proposition has consisted centred on the half-normal distribution, the essence of its methodological proposition has consisted of developing CPCDs through a right-truncated probability distribution applied on a hazards rate. of developing CPCDs through a right-truncated probability distribution applied on a hazards rate. TheThehalf-normal half-normal distribution distributionbrings bringsthetheadvantage advantageof of depending depending only only on on its itsscale parameter( σ scaleparameter (σ), but at ), but theatsame the same time, this property limits its capacity for studying scenarios that would be better modelled by time, this property limits its capacity for studying scenarios that would be better modelled changing the shape by changing of the the shape distribution. of the distribution.The Thegeneralised generalisedhalf-normal distribution(GHN), half-normal distribution (GHN), proposed proposed bybyCooray Cooray and andAnanda Ananda[11],[11],incorporates incorporates aa shape parametertotothe shape parameter theoriginal originalhalf-normal half-normal distribution, distribution, adding flexibility to its analytical capacity. This section explores the main changes that adding flexibility to its analytical capacity. This section explores the main changes that switching switching from thefrom half-normal to its generalised the half-normal counterpart to its generalised producesproduces counterpart on the previous results ofresults on the previous this paper, focusing of this paper,on HIaR mainly.onThe focusing probability HIaR mainly.density function of the The probability generalised density function half-normal distributionhalf-normal of the generalised ([11], p. 1125) distribution is as follows: ([11], p. 1125) is as follows: r    2 α x α − 1 ( x )2α 2α GHN = eα 2 1σ x  (29) 2 π αx  σx  − 2  σ  GHN =    e (29) if x > 0 and 0 otherwise. π  x  σ  In (29), α stands for the shape parameter and σ stands for the location parameter, substituting the if x > 0 and 0 otherwise. Greek letter theta (θ) used by Cooray and Ananda on behalf of homogeneity with the half-normal In (29), α stands for the shape parameter and σ stands for the location parameter, distribution notations previously used in this paper and in other papers quoted in the Section 1 and substituting the Greek letter theta (θ ) used by Cooray and Ananda on behalf of homogeneity with in the Appendix. The properties of the generalised half-normal distribution, including its statistical the half-normal moments, distribution are presented notations by Cooray and previously Ananda [11].used in this paper and in other papers quoted in theNext, Section 1 and in the Appendix. The properties we focus on the applicability of the generalised of the half-normal generalised half-normal to HIaR. Thedistribution, basic results including its statistical moments, are presented by Cooray and Ananda [11]. for EHD are presented in Appendix B. The HIaR rate for the generalised half-normal (λ) fulfils the Next, we focus on the applicability of the generalised half-normal to HIaR. The basic results for following condition: EHD are presented in Appendix B.ZλThe− HIaR 2 forx the generalised half-normal λ q  α 1 x 2α rate e 2(σ) πα σ ( ) fulfils the following condition: dx = c f (30) x 0 Mathematics 2020, 8, 2080 17 of 27 which parallels (4) for the original half-normal. Calculating this integral and solving it for λ, we obtain the values of the HIaR rate for σ, c f , and α: √  α1 λ=σ 2 · Er f −1 [c f ] (31) Taking (10) into account, this equation can be written as follows: 1 λ = σ · γ1,c α f (32) As for the notations, λ denotes the generalised HIaR rate for the generalised half-normal distribution, henceforth generalised HIaR rate, while γ continues designating the same rate for the ordinary half-normal distribution. Both rates become equal for α = 1 (Pescim et al. [12], p. 946). The parameters σ, c f , and α remain implicit in λ, the complete notation of which would be λσ,c f ,α . The generalised HIaR rate, as its ordinary counterpart, embeds the severity effect and the probability effect. The former (σ) remains unchanged. The latter equates the HIaR rate for σ = 1 powered at the inverse of the shape parameter α. The comparison between the probability effects of the generalised and the ordinary half-normal distribution explains the changes that the shape parameter introduces in the generalised HIaR rate. Table 2 summarises these changes. h √ i The first step of this comparison consists of realising that a confidence level, c f , equal to Er f 1/ 2 (0.6827 approximately) Mathematics 2020, 8, x FOR PEER REVIEW h √ i 18 of 29 neutralises the α impact because it equates γ1,c f to 1. When c f < Er f 1/ 2 and, thus, γ1 < 1, a shape h √ i parameter lower than 1 increases the probability effect, while c f >λEr f 1/γ 2 increases it because now Table 2. Comparing h √ theiprobability effects ( 1 vs. 1 ). we have γ1 > 1. The effects for c f > Er f 1/ 2 are the opposite. 0 < cf < Erf 1 / 2  cf = Erf 1 / 2  cf > Erf 1 / 2  Table 2. Comparing the probability effects (λ 1 vs. γ1 ). γ1,cf <1 √ γ1,cf = 1√ γ >1 1,cf √ 0Erf[1/ 2] α <1 λ < γ γ1,cf <1 λ =γγ1,cf==11 λ >1 γ1,cf >γ α =1 α<1 λ =γ λ<γ λ=γ=1 λ =γ =1 λ>γ λ =γ α >1 α=1 λ >γ λ=γ λ λ==γ γ==11 λ=λ γ< γ α>1 λ>γ λ=γ=1 λ<γ Region Region plot: plot: thethe blue blue zones zones λ λ<<γγ,, and denote denote and the white zones denote λλ>>γ.γ . the white (γ ) Table 2 compares HIaR rates for the half-normal distribution (γ) and the generalised half-normal distribution (λ). Table 2 compares HIaR rates for the half-normal distribution and the generalised half-normal Since both are directly proportional to the severity effect σ , comparing λ and γ is equivalent to comparing λ1,c f ,α ( ) distribution (λ ) in white the zones that. Since both fulfil λ are directly proportional to the severity effect 1 > γ1 . , comparing (σ ) and γ1,c f . The region plot, built through the Mathematica software, highlights in blue the zones in which λ1 < γ1 and λ and γ is equivalent to comparing λ1,cf ,α and γ 1,cf . The region plot, built through the Mathematica software, highlights in blue the zones in which λ1 < γ 1 and in white the zones that fulfil λ1 > γ 1 . The study of the ratio between the HIaR rates for the generalised and the ordinary half-normal distribution, henceforth denoted by ς , enlightens the role of the shape parameter. The expression of this ratio is as follows: 1  −1 Mathematics 2020, 8, 2080 18 of 27 The study of the ratio between the HIaR rates for the generalised and the ordinary half-normal distribution, henceforth denoted by ς, enlightens the role of the shape parameter. The expression of this ratio is as follows: λ √ ( 1 −1) ς= = 2 · Er f −1 [c f ] α (33) γ which, in turn, consists of the ratio of both probability effects, that, recalling (10) and (32), can be written as follows: ( 1 −1) ς = γ1,cα f (34) Thus, (ς − 1) expresses the percentage in which α changes the probability effect and, at the same time, the HIaR rate. Interestingly, high α values compensate the low probability effect associated with very low probabilities. Often, academic papers on risk analysis have pointed out that scenarios with high severity and low probability should be rated riskier than scenarios with lower severity and higher probability that present the same value of the risk indicator. In this respect, Duijim [21] (p. 27) defined the hazard aversion as “the attitude that a low probability-large consequence event is assigned a higher risk value than a high probability-low consequence event, even when the expected loss for both events is the same”. The capacity of the shape parameter of the generalised half-normal distribution for weighting the probability effect opens the way for increasing its impact of HIaR and EHD in case of low probabilities by assigning to the shape parameter the function of a hazard aversion coefficient in the case of low probabilities and high impacts. The development of this topic needs further research. Table 3 displays the values of ς for different values of α (columns) and c f (rows). Among other data, it shows that, for a 5% probability (cf), α equal to 2 increases the probability effect and the HIaR ratio at the same time, multiplying it by 3.993. Table 3. HIaR rates ratio (λ/γ). α 0.1 0.5 1 1.5 2 cf 0.05 0.0000 0.0627 1 2.5171 3.9934 0.5 0.0289 0.6745 1 1.1403 1.2176 0.6827 1 1 1 1 1 0.75 3.5275 1.1503 1 0.9544 0.9324 0.99 4991.76 2.5758 1 0.7295 0.6231 Table 3 displays the values of the HIaR rates ratio for the given values of the shape parameter (α) and the confidence level (cf ). Figure 9 represents the evolution of the generalised HIaR ratio for σ = 1, which is, at the same time, the evolution of the probability effect for the generalised h √ HIaR i ratio. The vertical plane is set up at α = 1, while the horizontal plane is set up at c f = Er f 1/ 2 (0.6827). The intersection between λ1 (yellow surface) and the vertical plane shows the line of γ1 because, as stated, λ1 equates γ1 for α = 1. The crossing between λ1 and the horizontal h √ plane i shows how λ1 evolves from values lower than 1 to values higher than 1 because c f = Er f 1/ 2 equates both γ1 and λ1 to 1. Beyond these applications, the generalised half-normal distribution offers an interesting potential for its use in the analysis of VaR, CVaR, and CPCDs when fat tails and skew become a central point in the analysis, paralleling the developments of the generalised normal distribution made, among others, by Stoyanov et al. [37] and Chen et al. [38].   λ1 to 1. Beyond these applications, the generalised half-normal distribution offers an interesting potential for its use in the analysis of VaR, CVaR, and CPCDs when fat tails and skew become a Mathematics central point 8, 2080 2020,in 19 of 27 the analysis, paralleling the developments of the generalised normal distribution made, among others, by Stoyanov et al. [37] and Chen et al. [38] Mathematics 2020, 8, x FOR PEER REVIEW 20 of 28 Figure 9. Generalised HIaR rate for σ = 1 (probability effect): Figure 9 represents the probability effect forFigure HIaR 9. Generalised under HIaR rate the generalised for σ =distribution. half-normal 1 (probability In effect): concrete, Figure these 9plots represents the probability show, from different effect for HIaR under the generalised half-normal distribution. In concrete, these points of view, the evolution of the generalised HIaR rate as a function of the confidence level plots show, from and the different shape pointsthat, parameter of view, the can as stated, evolution of the generalised be interpreted HIaR rate as a hazard aversion as a function coefficient. of the confidence The two-dimensional level and the shape parameter that, as stated, can be plot also represents the probability effect having parametrisedinterpreted as a hazard α at 0.5 (green line), 1 (bluecoefficient. aversion The line), and 1.5 two-dimensional plot also represents the probability n effect √ o having (red line). This plot emphasises the intersection point at Er f [1/ 2, 1 . parametrised α at 0.5 (green line), 1 (blue line), and 1.5 (red line). This plot emphasises the intersection point at 9. Discussion {Erf [1 / } 2,1 . Truncated probability distributions embed the essential features for analysing risky scenarios 9. Discussion that may produce only negative impacts on the assets related to them. Among these distributions, Truncated turns the half-normal probability out todistributions embed the essential be the straightforward features option for for analysing adapting VaR and risky CVaRscenarios to the that may produce only probability–severity negativeperformed risk analysis impacts on the assets through related CPCDs, to them. which Among overcome the these distributions, misclassifications thetohalf-normal due turnsstructure the rectangular out to beof the riskstraightforward option matrices. In effect, the for adapting iso-risk VaR separate lines that and CVaRtheto the risk probability–severity risk analysis performed through CPCDs, which overcome the misclassifications categories in CPCDs open the way for expressing them through risk indicators based on VaR and due to the rectangular structure of risk matrices. In effect, the iso-risk lines that separate the risk categories in CPCDs open the way for expressing them through risk indicators based on VaR and CVaR. Switching from the product between probability and severity to VaR and CVaR heightens the information level generated by the analysis. At the same time, it requires adapting VaR and CVaR to a setting characterised by truncated probability distributions. We have called the Truncated Risk (TR) Mathematics 2020, 8, 2080 20 of 27 CVaR. Switching from the product between probability and severity to VaR and CVaR heightens the information level generated by the analysis. At the same time, it requires adapting VaR and CVaR to a setting characterised by truncated probability distributions. We have called the Truncated Risk (TR) model the outcome of adapting VaR and CVaR to CPCDs. The methodology of the TR model has departed from defining the safety index and the hazards index analysed through the half-normal distribution. The former is referred to as a generic asset and represents the percentage of not being damaged by hazards. The latter expresses the impact of hazards on the safety index. Since the hazards index only captures negative impacts, although expressed in a positive sign, its random behaviour must be expressed through a truncated probability distribution. In this paper, we have opted for the half-normal distribution as the most straightforward issue. As a risk analysis tool, the half-normal distribution shows the advantage of depending on its standard deviation exclusively, namely, its scale parameter. Thus, for any risk setting that can be represented by a half-normal distribution reasonably, the standard deviation can be taken as an indicator of the severity of hazards. The probability of each hazard is now expressed through the maximum expected probability of the hazard occurrence, which turns out to be the confidence level associated with this probability. As a logical outcome, the probability on the right-hand side of the expected one becomes the surprise probability. It is, in other words, the probability that, in subjective terms, expresses the degree of the analyst believing that the hazard will not take place. The generalised half-normal distribution amplifies the analysis by including a shape parameter that adds flexibility to the model. As shown, the two central risk indicators of the TR model, HIaR and EHD, can be interpreted as the product between a probability effect and a severity effect. Thus, they hold the original message of risk matrices. The comparison between both approaches enlightens the changes in the probability–severity analysis proposed in this paper. Instead of obtaining the estimated total impact (ETI) as a direct product between probability (p) and impact severity (IS), the TR model filters the information through the half-normal distribution. Now, this distribution embeds all possible impacts and their probabilities. The volatility drives the severity of the impacts because it determines the shape of the half-normal distribution. The probability of the traditional approach is replaced by the confidence level or the half-normal distribution. The HIaR shares with the traditional impact size (IS) the fact of expressing the maximum damage expected by the analyst because it expresses the maximum expected impact under the maximum expected probability. The primary difference between the traditional impact size and HIaR lies in the fact that the traditional impact size is a unique value associated with the hazard in any circumstance in which the hazard becomes real. At the same time, HIaR is based on a probability distribution, not necessarily the half-normal. EHD is the concept that replaces the traditional total expected impact. In this way, the analysis gains flexibility because the single value of the estimated impact is now replaced by a tripled of values, which includes the total possible mathematical expectation (TME), the EHD, and its complement: the CHIaR. Thus, this approach induces decision-makers to make plans to face not only the expected (HIaR and EHD) but also the unexpected through CHIaR. As a clarification, it is worth pointing out that EHD does not have an equivalent measure in the framework of VaR analysis. HIaR parallels VaR, while CHIaR parallels CVaR, but the mathematical expectation of potential losses lower than VaR is not a standard risk indicator in financial analysis. Nevertheless, in the context of risk analysis through the half-normal distribution, EHD has become central for linking the VaR analysis developed in this paper with the traditional probability–severity approach of risk matrices. Section 7 has presented a subjective, although analytical, approach to the TR model that stresses its side as a thinking tool for enabling analysts to find the risk parameters hidden on their subjective expectations on downside risk. The extension of the analysis to the generalised half-normal distribution has shown the property of the shape coefficient in modifying the probability effect increasing the weight of lower probabilities and decreasing the weight of the higher ones or in producing the opposite effect, depending on the choice of the shape parameter. The properties of the shape parameter justify interpreting it as the hazard aversion coefficient. Risk aversion and hazard aversion are different Mathematics 2020, 8, 2080 21 of 27 indicators: the former determines the required risk premium, while the latter rules by assigning risk levels in the combinations of the probability and severity of different hazards. It is, in particular, relevant for assigning higher risk levels to hazards of high impact and low probability when they are compared with low impact and high probability hazards. Besides, the introduction of the generalised half-normal distribution has shown how the TR model can be adapted to a truncated probability distribution different from the half-normal that has guided its development. This paper has the limitation of having centred the analysis in the half-normal distribution. Its extension to the generalised half-normal has shown how more complex probability distributions may incorporate additional features than the ones of the half-normal. The risk measures that this paper has adapted to the probability–severity analysis, VaR and CVaR, have been extended also in the financial mathematics literature from the normal distribution to other probability distributions that may fit better to specific cases, fat tails in particular. These extensions take also into account the limitations of VaR in the face of subadditivity. In addition, interweaving the point of view of truncated distributions with some of the advances in risk management mentioned in the Introduction section, such as considering the knowledge dimension of risk, may contribute as well to widening the scope of the analysis presented in this paper. 10. Conclusions By applying the half-normal distribution as its primary methodological tool, this paper has adapted VaR and CVaR to probability–severity risk analysis by building up a model that we have called the Truncated Risk (TR) model. The two central risk indicators proposed, HIaR and EHD, can be broken down into a severity effect that consists of the risk rate volatility and a probability effect that has the confidence level as its unique independent variable. Both indicators have been related functionally to compare and strengthen their meanings. The EHD truncates at the HIaR point the half-normal distribution that stems from the severity effect expressed by the risk rate volatility, which is, at the same time, the location parameter of the distribution. The part of the half-normal distribution not absorbed by the EHD, i.e., the one on the right-hand side of HIaR, can be interpreted as the unexpected hazards impact, which means the expected hazards damage embedded in the severity parameter but excluded by the estimation of the maximum probability of the hazard occurrence, namely, the confidence level, in the TR model. The division of the total mathematical expectation generated by the severity coefficient between EHD and CHIaR may help to consider surprises in addition to expectations in risk analysis. A simplified model, based on subjective probabilities, shows how the TR model may contribute to turning generic risk appreciations into a set of related parameters, the coherence of which can be revised by decision-makers to improve the outcomes of subjective appreciations. The extension of the TR model to the generalised half-normal distribution has shown how incorporating a shape parameter enlarges the analytical capacity of the original approach. However, the most relevant conclusion of the extension to the generalised half-normal distribution has been to realise how the basic TR model can be broadened by introducing other truncated probability distributions that may fit better with the scenarios under analysis. Scheme A1 summarises the contents of this paper in a conceptual map. Author Contributions: Conceptualization, M.-T.B.-B., J.M.-S. and M.-A.T.-R.; Formal analysis, M.-T.B.-B., J.M.-S. and M.-A.T.-R.; Methodology, M.-T.B.-B., J.M.-S. and M.-A.T.-R.; Writing—original draft, M.-T.B.-B., J.M.-S. and M.-A.T.-R.; Writing—review–editing, M.-T.B.-B., J.M.-S. and M.-A.T.-R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. Mathematics 2020, 8, 2080 22 of 27 Appendix A. Half-Normal Distribution This appendix summarises the main properties of the half-normal distribution used in this paper. The half-normal distribution is a folded normal distribution with zero mean. Its probability density function (Eland [39], p.553) is as follows: √ 1 2 −x2 /2σ2 f (x) = √ e (A1) σ π where σ stands for the standard deviation of the normal distribution that, once folded, is turned into the half-normal. Besides, x ≥ 0. Being more specific, the half-normal distribution is generated by adding the condition x = |z| to a normal variable z distributed as N (0, σ). Often (Weisstein [40]), the half-normal probability density function is expressed by introducing the parameter: √ 1 π θ= √ (A2) σ 2 Then, 2 −x2 θ2 /π f (x) = θe (A3) σ Elandt ([39], p. 552) obtained the first fourth moments of the half-normal distribution. Let us point out that the symbol θ has a different meaning in [39,40]. In this appendix, we use the one of [40]. Leone et al. [41] also studied the moments of the half-normal distribution. Johnson [42] approached the estimation of the standard errors. For an updated and extensive analysis of the half-normal distribution, see [43]. In the TR model, we make use of the links between the mean (xHN ) and the standard deviation (σHN ) of the half-normal distribution with the standard deviation of unfolded normal distribution (σ). Their equations are as follows: √ 2 xHN = σ √ (A4) π r 2 σHN = 1 − σ (A5) π When introducing θ, these equations become 1 xHN = (A6) θ r 1 π−2 σHN = (A7) θ 2 The software employed in this paper, Mathematica, uses the parameter θ in its presentation of the half-normal distribution [40]. The generalisation of the half-normal distribution proposed by Cooray and Ananda [11] consists of adding a shape parameter to the half-normal distribution, leading to the probability density function shown in (29). Figure A1 compares the shape of the half-normal distribution with different shapes of its equivalent counterpart. The former is equivalent to the latter for a shape parameter equal to 1. The role of the shape parameter, related to HIaR, has been studied in Section 8. shapes of its equivalent counterpart. The former is equivalent to the latter for a shape parameter equal to 1. The role of the shape parameter, related to HIaR, has been studied in Section 8. Mathematics 2020, 8, 2080 23 of 27 σ = 1 α ={0.5, 1, 4} (Green-Blue-Red) σ = 0.5 α={0.5, 1, 4} (Green-Blue-Red) Mathematics 2020, 8, x FOR PEER REVIEW 24 of 28 σ = 2 α={0.5, 1, 4} (Green-Blue-Red) Figure Generalisedhalf-normal Figure A1. Generalised half-normaldistribution: distribution: Figure Figure A1 A1 shows shows the the plotsplots of generalised of the the generalised half- half-normal distribution normal distribution for thefor the values given given values of the standard of the standard deviationdeviation and the and the shape shape parameter. parameter. The shape The shape parameter parameter equal to 1 equal to 1 (blue (blue lines) lines) corresponds corresponds to thehalf-normal to the ordinary ordinary half-normal distribution. distribution. The The analysis analysis developed developed in in the the text text of of the the article article departs departs directly directly from from the the probability probability density density function shown in (29). Cooray and Ananda [11] present the full details function shown in (29). Cooray and Ananda [11] present the full details of this distribution. of this distribution. Several Several extensionsextensions of the generalised of the generalised half-normal half-normal distribution distribution have beenhave been proposed, proposed, among among others, others, by Pescim by Pescim et al. [12], Olmos et al. [44], Duarte Sanchez et al. [45], et al. [12], Olmos et al. [44], Duarte Sanchez et al. [45], and Ascitas [46]. and Ascitas [46]. Appendix B: The Expected Hazards Damage through the GHND The Expected Hazards Damage (EHD) for the generalised half-normal distribution is obtained by substituting in (12) the half-normal distribution for its generalised counterpart. Referring EHD to the unitary HIaR, we can write the following: 2α λ 1 x  α 1 −   2  α  x  EHD = x ⋅ e 2  σ   dx π  x  (A8) cf 0  σ  Mathematics 2020, 8, 2080 24 of 27 Appendix B. The Expected Hazards Damage through the GHND The Expected Hazards Damage (EHD) for the generalised half-normal distribution is obtained by substituting in (12) the half-normal distribution for its generalised counterpart. Referring EHD to the unitary HIaR, we can write the following: Zλ r    1 − 12 ( σx )2α 2 α x α EHD = x·e dx (A8) cf π x σ 0 Solving the integral in Equation (A8), we obtain the following: 1 1 2 2 /α 1+α 1 + α 1 λ 2α   "   #! EHD = √ σ Γ −Γ , (A9) cf π 2α 2α 2 σ 1+α h i We find that EHD now depends on the Gamma function Γ 2α and on the incomplete Gamma     1+α 1 λ 2α function Γ , 2α 2020, Mathematics , which σ x FOR 2 8, must be numerically evaluated. PEER REVIEW 26 of 29 Figure A2 shows the plots of EHD for the generalised half-normal distribution. Figure A2 shows the plots of EHD for the generalised half-normal distribution. α = 0.5 α = 1.5 Figure A2. EHD Figure for the A2. EHD generalised for the generalisedhalf-normal half-normal distribution: theupper distribution: the upperrowrow of this of this figure figure displays displays two continuous probability–consequence two continuous probability–consequencediagrams (CPCDs)for diagrams (CPCDs) forshape shape parameters parameters equalequal toand to 0.5 0.5 and 1.5. The 1.5. bottom rowrow The bottom shows showsthethe correspoding correspodingthree-dimensional plots three-dimensional plots ofof thethe EHD EHD function function withwith the the standard deviation standard andand deviation thethe confidence confidencelevel levelasasthe theindependent variables. independent variables. Mathematics 2020, 8, 2080 25 of 27 Mathematics 2020, 8, x FOR PEER REVIEW 26 of 28 Scheme Scheme A1. Conceptual map. A1. Conceptual map. References References 1. 1. Aharony, J.; Aharony, J.; Jones, Jones, C.P.; C.P.; Swary, Swary,I.I. An An Analysis Analysis of of Risk Risk and and Return Return Characteristics Characteristics of of Corporate Bankruptcy Corporate Bankruptcy Using Capital Market Data. J. Financ. 1980, 35, 1001–1016. [CrossRef] Using Capital Market Data. J. Financ. 1980, 35, 1001–1016, doi:10.1111/j.1540-6261.1980.tb03516.x. 2. 2. Collins, R.A.; Collins, R.A.; Gbur, Gbur,E.E. E.E.Risk RiskAnalysis Analysis forfor Proprietors Proprietors with Limited with Liability: Limited A Mean-Variance, Liability: A Mean-Variance, Safety-First Safety- Synthesis. West. J. Agric. Econ. 1991, 16, 156–162. First Synthesis. West. J. Agric. Econ. 1991, 16, 156–162. 3. 3. Ergashev, B.; Ergashev, B.;Pavlikov, Pavlikov,K.;K.;Uryasev, Uryasev, S.;S.; Sekeris, Sekeris,E. Estimation of Truncated E. Estimation DataData of Truncated Samples in Operational Samples Risk in Operational Modeling. J. Risk Insur. 2015, 83, 613–640. [CrossRef] Risk Modeling. J. Risk Insur. 2015, 83, 613–640, doi:10.1111/jori.12062. 4. 4. Chen, C.W.S.; Chen, C.W.S.; Gerlach, R.; Lin, Gerlach, R.; Lin, E.M.H.; E.M.H.; Lee, Lee, W.C.W. W.C.W. Bayesian Bayesian Forecasting for Financial Forecasting for Financial Risk Risk Management, Management, Pre and Post the Global Financial Crisis. J. Forecast. 2011, 31, 661–687. [CrossRef] Pre and Post the Global Financial Crisis. J. Forecast. 2011, 31, 661–687, doi:10.1002/for.1237. 5. 5. De Roon, De Roon,F.; F.;Karehnke, Karehnke,P.P.AASimple Simple Skewed Skewed Distribution Distribution with with Asset Asset Pricing Pricing Applications. Applications. Rev. Financ. Rev. Financ. 2016, 2016, rfw040. [CrossRef] rfw040, doi:10.1093/rof/rfw040. 6. 6. Jawitz, J.W. Jawitz, J.W.Moments Momentsofoftruncated truncatedcontinuous continuousunivariate univariatedistributions. distributions.Adv. Water Adv. Resour. Water 2004, Resour. 27, 27, 2004, 269–281. 269– 7. Li, J.; Thyer, M.; Lambert, M.; Kuczera, G.; Metcalfe, A. An efficient causative event-based approach for 281. deriving the annual flood frequency distribution. J. Hydrol. 2014, 510, 412–423. 7. Li, J.; Thyer, M.; Lambert, M.; Kuczera, G.; Metcalfe, A. An efficient causative event-based approach for 8. Lazar, N.; Dolšek, M. A closed form solution for seismic risk assessment incorporating intensity bounds. deriving the annual flood frequency distribution. J. Hydrol. 2014, 510, 412–423. Eng. Struct. 2014, 78–89. [CrossRef] 8. Lazar, N.; Dolšek, M. A closed form solution for seismic risk assessment incorporating intensity bounds. 9. Krenek, R.; Cha, J.; Cho, B.R. Development of the convolutions of truncated normal random variables with Eng. Struct. 2014, 78–89, doi:10.1016/j.engstruct.2014.07.01. three different quality characteristics in engineering applications. Comput. Ind. Eng. 2016, 94, 125–137. 9. Krenek, R.; Cha, J.; Cho, B.R. Development of the convolutions of truncated normal random variables with [CrossRef] three different quality characteristics in engineering applications. Comput. Ind. Eng. 2016, 94, 125–137, 10. Krause, A.L.; Kurowski, L.; Yawar, K.; Van Gorder, R.A. Stochastic epidemic metapopulation models on doi:10.1016/j.cie.2015.12.014. networks: SIS dynamics and control strategies. J. Theor. Biol. 2018, 449, 35–52. [CrossRef] 10. Krause, A.L.; Kurowski, L.; Yawar, K.; Van Gorder, R.A. Stochastic epidemic metapopulation models on 11. Cooray, K.; Ananda, M.M.A. A Generalization of the Half-Normal Distribution with Applications to Lifetime networks: SIS dynamics and control strategies. J. Theor. Biol. 2018, 449, 35–52, doi:10.1016/j.jtbi.2018.04.023. Data. Commun. Stat. 2008, 37, 1323–1337. [CrossRef] 11. Cooray, K.; Ananda, M.M.A. A Generalization of the Half-Normal Distribution with Applications to Lifetime Data. Commun. Stat. 2008, 37, 1323–1337, doi:10.1080/03610920701826088. Mathematics 2020, 8, 2080 26 of 27 12. Pescim, R.R.; Demétrio, C.G.B.; Cordeiro, G.M.; Ortega, E.M.M.; Urbano, M.R. The beta generalised half-normal distribution. Comput. Stat. Data Anal. 2010, 54, 945–957. [CrossRef] 13. Hopkin, P. Fundamentals of Risk Management: Understanding, Evaluating and Implementing Effective Risk Management; Kogan Page Publishers: London, UK, 2018. 14. Anthony (Tony) Cox, L., Jr. What’s wrong with risk matrices? Risk Anal. Int. J. 2008, 28, 497–512. 15. Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D. Coherent Measures of Risk. Math. Financ. 1999, 9, 203–228. [CrossRef] 16. Li, J.; Bao, C.; Wu, D. How to Design Rating Schemes of Risk Matrices: A Sequential Updating Approach. Risk Anal. 2017, 38, 99–117. [CrossRef] [PubMed] 17. Aven, T. Risk Analysis; John Wiley & Sons: Chichester, West Sussex, UK, 2015. 18. Aven, T. Improving risk characterisations in practical situations by highlighting knowledge aspects, with applications to risk matrices. Reliab. Eng. Syst. Saf. 2017, 167, 42–48. [CrossRef] 19. Levine, E.S. Improving risk matrices: The advantages of logarithmically scaled axes. J. Risk Res. 2012, 15, 209–222. 20. Baybutt, P. Calibration of risk matrices for process safety. J. Loss Prev. Process Ind. 2015, 38, 163–168. [CrossRef] 21. Duijm, N.J. Recommendations on the use and design of risk matrices. Saf. Sci. 2015, 76, 21–31. [CrossRef] 22. Ale, B.; Burnap, P.; Slater, D. On the origin of PCDS—(Probability-consequence diagrams). Saf. Sci. 2015, 72, 229–239. [CrossRef] 23. Ni, H.; Chen, A.; Chen, N. Some extensions on risk matrix approach. Saf. Sci. 2010, 48, 1269–1278. [CrossRef] 24. Laine, V.; Valdez Banda, O.A.; Goerlandt, F. Event risk classification method for pollution preparedness and response. In Developments in the Collision and Grounding of Ships and Offshore Structures; CRC Press: London, UK, 2019; Volume 11, pp. 344–354. [CrossRef] 25. Hull, J. Risk Management and Financial Institutions; John Wiley & Sons: Hoboken, NJ, USA, 2018. 26. Alexander, C. Value-at-Risk Models; Wiley: New York, NY, USA, 2008. 27. Hubbert, S. Essential Mathematics for Market Risk Management; John Wiley & Sons: Chichester, West Sussex, UK, 2011; Volume 642. 28. Daníelsson, J.; Jorgensen, B.N.; Samorodnitsky, G.; Sarma, M.; de Vries, C.G. Fat tails, VaR and subadditivity. J. Econom. 2013, 172, 283–291. [CrossRef] 29. Karni, E. Axiomatic Foundation of Expected Utility and Subjective Probability. In Handbook of the Economics of Risk and Uncertainty; Machina, M., Viscusi, K., Eds.; Elsevier: Amsterdam, The Netherlands, 2014; pp. 1–39. [CrossRef] 30. Andersen, S.; Fountain, J.; Harrison, G.W.; Rutström, E.E. Estimating subjective probabilities. J. Risk Uncertain. 2014, 48, 207–229. [CrossRef] 31. Wintle, B.C.; Fraser, H.; Wills, B.C.; Nicholson, A.E.; Fidler, F. Verbal probabilities: Very likely to be somewhat more confusing than numbers. PLoS ONE 2019, 14, e0213522. [CrossRef] 32. Aven, T.; Reniers, G. How to define and interpret a probability in a risk and safety setting. Saf. Sci. 2013, 51, 223–231. [CrossRef] 33. Goerlandt, F.; Reniers, G. On the assessment of uncertainty in risk diagrams. Saf. Sci. 2016, 84, 67–77. [CrossRef] 34. Flage, R.; Aven, T.; Berner, C.L. A comparison between a probability bounds analysis and a subjective probability approach to express epistemic uncertainties in a risk assessment context—A simple illustrative example. Reliab. Eng. Syst. Saf. 2018, 169, 1–10. [CrossRef] 35. Langdalen, H.; Abrahamsen, E.B.; Abrahamsen, H.B. A New Framework to Identify and Assess Hidden Assumptions in the Background Knowledge of a Risk Assessment. Reliab. Eng. Syst. Saf. 2020, 200, 106909. [CrossRef] 36. Cox, J.C.; Ross, S.A.; Rubinstein, M. Option pricing: A simplified approach. J. Financ. Econ. 1979, 7, 229–263. [CrossRef] 37. Stoyanov, S.V.; Rachev, S.T.; Fabozzi, F.J. CVaR sensitivity with respect to tail thickness. J. Bank. Financ. 2013, 37, 977–988. [CrossRef] 38. Chen, C.R.; Su, Y.; Huang, Y. Hourly index return autocorrelation and conditional volatility in an EAR–GJR-GARCH model with generalised error distribution. J. Empir. Financ. 2008, 15, 789–798. [CrossRef] Mathematics 2020, 8, 2080 27 of 27 39. Elandt, R.C. The Folded Normal Distribution: Two Methods of Estimating Parameters from Moments. Technometrics 1961, 3, 551–562. [CrossRef] 40. Weisstein, E.W. Half-Normal Distribution. From MathWorld—A Wolfram Web Resource. Available online: https://mathworld.wolfram.com/Half-NormalDistribution.html (accessed on 1 September 2020). 41. Leone, F.C.; Nelson, L.S.; Nottingham, R.B. The Folded Normal Distribution. Technometrics 1961, 3, 543–550. [CrossRef] 42. Johnson, N.L. The Folded Normal Distribution: Accuracy of Estimation By Maximum Likelihood. Technometrics 1962, 4, 249–256. [CrossRef] 43. Tsagris, M.; Beneki, C.; Hassani, H. On the Folded Normal Distribution. Mathematics 2014, 2, 12–28. [CrossRef] 44. Olmos, N.M.; Varela, H.; Bolfarine, H.; Gómez, H.W. An extension of the generalised half-normal distribution. Stat. Pap. 2013, 55, 967–981. [CrossRef] 45. Duarte Sanchez, J.J.; da Luz Freitas, W.W.; Cordeiro, G.M. The extended generalised half-normal distribution. Braz. J. Probab. Stat. 2016, 30, 366–384. [CrossRef] 46. Acitas, S. A new weighted distribution as an extension of the generalised half-normal distribution with applications. J. Stat. Comput. Simul. 2018, 88, 2325–2341. [CrossRef] Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).